<p> TESLA data </p>

51cb0b35-9690-4b43-8541-1664513e86c3 20 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

Solving a determined system of Non-linear equations

vector of guesses X0 0.01 1000 0.001 50021217 mat transpose :

AlgLib Solver for a system of nonlinear equations(Uni) sol k 1 β f rows range u k el β k el : for β Jac u f u Jacobian : StepMax 0 : ε 1014 -^: 12 mat u X0 StepMax ε β f β Jac al_nleqsolve : 41 line :

3 appVersion 0.98.6179 ≡

user scale factor m 0.001 :

0.01 1000 0.001 50021217 mat

Ua b β ia β 3 el 1 β 2 el / Ua * 1 β 2 el β 1 el β 6 el b + Ua β 5 el^β 4 el + β 5 el nthroot / + (* exp + ln * (β 7 el^* :

Solver function β f 400 β ia 1.044 m * - 1200 β ia 2.679 m * - 2100 β ia 4.749 m * - 20 0.5 - β ia 0.18 m * - 120 0.5 - β ia 1.66 m * - 240 0.5 - β ia 4.3 m * - 401 - β ia 0.132 m * - 71 mat :

sol 0.009993 999.999999 0.000866 500.000105 1.968166 0.83653 1.588528 71 mat

<p> TESLA data </p>

a1 0010 0.227 20 0.537 30 0.796 40 1.044 50 1.279 60 1.495 70 1.698 80 1.894 90 2.085 100 2.278 110 2.476 120 2.679 130 2.889 140 3.103 150 3.324 160 3.549 170 3.779 180 4.015 190 4.225 200 4.5 210 4.749 222 mat :

b1 0010 0.088 20 0.18 30 0.277 40 0.393 50 0.5 60 0.631 70 0.776 80 0.932 90 1.1 100 1.28 110 1.46 120 1.66 130 1.86 140 2.07 150 2.28 160 2.5 170 2.71 180 2.93 190 3.15 200 3.38 210 3.6 220 3.84 230 4.07 240 4.3 252 mat :

c1 0010 0.031 20 0.063 30 0.096 40 0.132 50 0.17 60 0.216 70 0.262 80 0.324 90 0.4 100 0.5 110 0.622 120 0.764 130 0.923 140 1.098 150 1.278 160 1.466 170 1.659 180 1.857 190 2.059 200 2.265 210 2.474 220 2.687 230 2.902 240 3.119 250 3.338 260 3.558 270 3.779 282 mat :

d1 0060 0.032 70 0.064 80 0.097 90 0.133 100 0.17 110 0.211 120 0.26 130 0.321 140 0.4 150 0.5 160 0.626 170 0.773 180 0.935 190 1.105 200 1.278 210 1.449 220 1.62 230 1.8 240 1.991 250 2.2 260 2.428 270 2.668 280 2.907 290 3.132 300 3.333 262 mat :

e1 00110 0.027 120 0.055 130 0.086 140 0.12 150 0.16 160 0.206 170 0.262 180 0.327 190 0.406 200 0.5 210 0.61 220 0.736 230 0.876 240 1.028 250 1.2 260 1.362 270 1.543 280 1.734 290 1.936 300 2.15 310 2.375 320 2.611 330 2.853 340 3.1 252 mat :

f1 00160 0.036 170 0.073 180 0.112 190 0.154 200 0.2 210 0.251 220 0.309 230 0.376 240 0.456 250 0.55 260 0.661 270 0.787 280 0.928 290 1.081 300 1.25 310 1.428 320 1.614 330 1.807 340 2.003 350 2.2 360 2.396 370 2.591 375 2.687 242 mat :

i1 00310 0.016 320 0.035 330 0.061 340 0.1 350 0.15 360 0.218 370 0.3 375 0.347 92 mat :

h1 00260 0.025 270 0.053 280 0.088 290 0.132 300 0.19 310 0.263 320 0.346 330 0.444 340 0.546 350 0.65 360 0.755 370 0.859 375 0.911 142 mat :

g1 00210 0.033 220 0.067 230 0.106 240 0.149 250 0.2 260 0.26 270 0.329 280 0.408 290 0.498 300 0.6 310 0.713 320 0.837 330 0.968 340 1.11 350 1.25 360 1.4 370 1.55 375 1.62 192 mat :

a1 a1 2 a1 rows range 12 range el : b1 b1 2 b1 rows range 12 range el : c1 c1 2 c1 rows range 12 range el : d1 d1 2 d1 rows range 12 range el : e1 e1 2 e1 rows range 12 range el : f1 f1 2 f1 rows range 12 range el : g1 g1 2 g1 rows range 12 range el : h1 h1 2 h1 rows range 12 range el : i1 i1 2 i1 rows range 12 range el : 91 sys

a1 b1 c1 d1 e1 f1 g1 h1 i1 19 mat

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 31 line for data r3 r4 r5 augment 21 line :

e1 11 mat 110 0.027 120 0.055 130 0.086 140 0.12 150 0.16 160 0.206 170 0.262 180 0.327 190 0.406 200 0.5 210 0.61 220 0.736 230 0.876 240 1.028 250 1.2 260 1.362 270 1.543 280 1.734 290 1.936 300 2.15 310 2.375 320 2.611 330 2.853 340 3.1 242 mat 11 mat

d1 2 col d1 1 col augment 11 mat 0.032 60 0.064 70 0.097 80 0.133 90 0.17 100 0.211 110 0.26 120 0.321 130 0.4 140 0.5 150 0.626 160 0.773 170 0.935 180 1.105 190 1.278 200 1.449 210 1.62 220 1.8 230 1.991 240 2.2 250 2.428 260 2.668 270 2.907 280 3.132 290 3.333 300252 mat 11 mat

Sanity wrt Mathcad β's ==============

j 1 :

sanity

TESLA a1 b1 c1 d1 e1 f1 g1 h1 i1 19 mat : tesla TESLA j el : XY tesla 1 col tesla 2 col m * augment : data XY o 5 black plot : 41 line

Triode voltage [V] Ua Aanode voltage (x) ≡ b Grid voltage ≡ 21 sys

recast the system model Ua b β ia β 3 el 1 β 2 el / Ua * 1 β 2 el β 1 el β 6 el b + β 4 el Ua 2^+ sqrt / + (* exp + ln * (β 7 el^* :

Mathcad Given/Find β 0.009728 783 0.001113 475.8 2 0.6425 1.49 17 mat transpose :

β 0.01 1000 0.000866 500 1.9682 0.83653 1.588528 17 mat transpose :

best fit TESLA [a1]
other TESLA [b1,c1 ...]
[d1,e1,f1,g1,h1,i1] 0.01 1000 0.000866 500 1.9682 0.83653 1.588528 17 mat 0.01 1000 0.000866 5002 0.8 1.601 17 mat 0.01 1000 0.000866 5002 0.76 1.62 17 mat 31 mat

Solver function x β nleq x 0 β ia x 0.5 - β ia x 1 - β ia x 1.5 - β ia x 2 - β ia x 2.5 - β ia x 3 - β ia x 3.5 - β ia x 4 - β ia 91 mat :

x β nleq 1 el x β nleq 2 el x β nleq 3 el x β nleq 4 el x β nleq 5 el x β nleq 6 el x β nleq 7 el x β nleq 8 el x β nleq 9 el data 10 1 sys

Conclude the project

1. Mathcad Given/Find
2. Smath optimized nleq 0.009728 783 0.001113 475.8 2 0.6425 1.49 17 mat 0.01 1000 0.000866 5002 0.8 1.601 17 mat 21 sys

β 3 el 1 β 2 el / Ua * 1 β 2 el β 1 el β 6 el b + Ua β 5 el^β 4 el + β 5 el nthroot / + (* exp + ln * (β 7 el^*

Final simplification
recast the system β 5 el 2 ≡ Ua β 5 el^β 4 el + β 5 el nthroot β 4 el Ua 2^+ sqrt ≡ 21 sys

β5 is now ignored for 
this particular project β 3 el 1 β 2 el / Ua * 1 β 2 el β 1 el β 6 el b + β 4 el Ua 2^+ sqrt / + (* exp + ln * (β 7 el^*

slope s 0.05 x * :

shift λ 1 :

'n 'curvature n 1 :

x n f s 1 x λ - (exp + ln n^(* :

j 4 :

sanity

m 0.001 :

x 1 f x 1 - ( n f x 2 - ( n f x 3 - ( n f x 4 - ( n f x 5 - ( n f 6 1 sys

TESLA a1 b1 c1 d1 e1 f1 g1 h1 i1 19 mat : tesla TESLA j el : XY tesla 1 col tesla 2 col m * augment : data XY o 5 black plot : 41 line

's*x' inciline
'λ' shift
'n' curvature s 0.000000019 x * : λ 60 : n 1.17 : 31 sys

x n f s 1 x λ - (exp + ln n^(* :

data x n f 2 1 sys